The

Hamiltonian Cycle Problem (HCP) is to identify a cycle in an undirected graph connecting all the vertices in the graph.

Ruskey and Savage in 1993 asked whether every matching in a hypercube can be extended to a

Hamiltonian cycle. A positive answer is known for perfect matchings, but the general case has been resolved only for matchings of linear size.

Applying Lemma 6, we conclude that H has a

Hamiltonian cycle C, and we obtain the desired result that V(D) = V([C.sub.3]) [union] V(C).

Speaking of a

Hamiltonian cycle system of order [upsilon], or HCS([upsilon]) for short, we mean a set of

Hamiltonian cycles of [K.sub.[upsilon]] whose edges partition E([K.sub.[upsilon]]) if [upsilon] is odd or E([K.sub.[upsilon]]) - I, with I a 1-factor of [K.sub.[upsilon]], if [upsilon] is even.

Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the puzzle that involves finding a

Hamiltonian cycle in the edge graph of the dodecahedron.

The midpoints of the edges of a

hamiltonian cycle in the 1-skeleton of a regular tetrahedron T are the vertices of a square.

The

hamiltonian cycle for the cases m = 10 and m = 11 are shown in the Figures 2.2 and 2.3 respectively.

If the degree sum of any two nonadjacent vertices is at least n, then G has a

Hamiltonian Cycle.

In this paper we prove that a through-vertex

Hamiltonian cycle exists in any triangular or tetrahedral grid under very mild conditions, and that there exist quadrilateral and hexahedral grids for which no unconstrained Hamiltonian path exists.

This special type of path is called a

Hamiltonian cycle. The prover's aim is to persuade a verifier that such a path is known without giving the verifier the slightest idea of how to construct the path.

A

Hamiltonian cycle of G is a cycle that traverses every vertex of G exactly once.

A trivial lower bound of 2 (for n [greater than or equal to] 4) is obtained from a minimum weight

Hamiltonian cycle in K(P), because this cycle is plane and consists of two edge-disjoint matchings.